Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion

Monotone capacities (on finite sets) of finite or infinite order (lower probabilities) are characterized by properties of their Mobius inverses. A necessary property of probabilities dominating a given capacity is demonstrated through the use of Gale’s theorem for the transshipment problem. This property is shown to be also sufficient if and only if the capacity is monotone of infinite order. A characterization of dominating probabilities specific to capacities of order 2 is also proved.

[1]  T. Fine,et al.  A Note on Undominated Lower Probabilities , 1986 .

[2]  Bernd Anger,et al.  Representation of capacities , 1977 .

[3]  Tatsuro Ichiishi,et al.  Super-modularity: Applications to convex games and to the greedy algorithm for LP , 1981 .

[4]  Isaac Levi,et al.  The Enterprise Of Knowledge , 1980 .

[5]  Jean-Yves Jaffray,et al.  Decision making in a case of mixed uncertainty: A normative model , 1985 .

[6]  R. E. Bixby,et al.  The Partial Order of a Polymatroid Extreme Point , 1985, Math. Oper. Res..

[7]  P. J. Huber,et al.  Minimax Tests and the Neyman-Pearson Lemma for Capacities , 1973 .

[8]  Bernd Anger,et al.  Approximation of capacities by measures , 1971 .

[9]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[10]  Glenn Shafer,et al.  Allocations of Probability , 1979, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[11]  J. Benzecri,et al.  Théorie des capacités , 1956 .

[12]  C. Dellacherie.  Quelques Commentaires Sur Les Prolongements De Cafacites , 1971 .

[13]  L. Shapley.  Cores of convex games , 1971 .

[14]  T. Fine,et al.  Varieties of modal (classificatory) and comparative probability , 1979, Synthese.

[15]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[16]  A. Revuz.  Fonctions croissantes et mesures sur les espaces topologiques ordonnés , 1956 .

[17]  T. Fine,et al.  Towards a Frequentist Theory of Upper and Lower Probability , 1982 .

[18]  Henry Ely Kyburg,et al.  The logical foundations of statistical inference , 1974 .

[19]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[20]  T. Fine,et al.  Bayes-like Decision Making With Upper and Lower Probabilities , 1982 .

[21]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[22]  David Gale.  The theory of linear economic models , 1960 .

[23]  Abraham Wald,et al.  Statistical Decision Functions , 1951 .